A293012 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x/(1 - x)^k).
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 13, 1, 1, 1, 7, 31, 73, 1, 1, 1, 9, 55, 241, 501, 1, 1, 1, 11, 85, 529, 2261, 4051, 1, 1, 1, 13, 121, 961, 6121, 24781, 37633, 1, 1, 1, 15, 163, 1561, 13041, 82711, 309835, 394353, 1, 1, 1, 17, 211, 2353, 24101, 207001, 1273567, 4342241, 4596553, 1
Offset: 0
Examples
E.g.f. of column k: A_k(x) = 1 + x/1! + (2*k + 1)*x^2/2! + (3*k^2 + 9*k + 1)*x^3/3! + (4*k^3 + 36*k^2 + 32*k + 1)*x^4/4! + ... Square array begins: 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, ... 1, 3, 5, 7, 9, 11, ... 1, 13, 31, 55, 85, 121, ... 1, 73, 241, 529, 961, 1561, ... 1, 501, 2261, 6121, 13041, 24101, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
Crossrefs
Programs
-
Mathematica
Table[Function[k, n! SeriesCoefficient[Exp[x/(1 - x)^k], {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten -
PARI
T(n, k) = n!*sum(j=0, n, binomial(n+(k-1)*j-1, n-j)/j!); \\ Seiichi Manyama, Mar 06 2023
Formula
E.g.f. of column k: exp(x/(1 - x)^k).
From Seiichi Manyama, Oct 21 2017: (Start)
Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.
A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*B(j,k)*A(n-j,k)/(n-j)! for n > 0. (End)
A(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*j-1,n-j)/j!. - Seiichi Manyama, Mar 06 2023